As we implement higher standards across the country, it has become increasingly important that we identify and use a variety of strategies to assess student learning so that the appropriate interventions may be provided. 

One strategy is to encourage students to reflect on their reasoning and justify their work.  The idea of justifying your work in mathematics has to go beyond the use of inverse operations to “prove” that the calculation was correct. This way of checking is not justification since it does not address the student’s use of metacognition—the thinking about thinking—that goes beyond the use of an algorithm and takes you into their decision-making processes.

We often use critical thinking and reflective thinking synonymously. Halpern describes critical thinking as “the use of cognitive skills or strategies that increase the probability of a desirable outcome … thinking that is purposeful, reasoned, and goal directed.”

Now if you think about that description of critical thinking, it really captures the essence of the type of thinking that is involved in solving problems, formulating inferences, calculating likelihoods, and making decisions in the mathematics classroom.

On the other hand, reflective thinking may be described as a part of the critical-thinking process.  It is when you analyze and make judgments about what has happened (past tense)—assessing what you know, what you need to know, and how to bridge that gap—during learning situations. This sounds just like the KWL model that we are so familiar with in teaching! Wow, now we see why it is so popular and effective.

We should focus on promoting reflections and reflective thinking  because it provides students with an opportunity to step back and think about how they actually solve problems and how a particular set of problem-solving strategies is appropriated for achieving their goal. By doing so, students will be better equipped to justify their answers because they can explain why they chose the operation, why they chose a particular strategy, and more importantly why it makes sense. The explanation of the algorithm would then be used to simply review the calculation for errors.

So, how do we create an environment where the development of reflective thinking may flourish?

Well, the answer is nothing new. It is all about the use of wait time. But I am learning that just waiting is not enough. I intentionally call this time “think time” and use hand signals to facilitate the think time and provide specific time frames for the thinking, along with a clearly focused task to think about. Sometimes the task will include a word bank that students use to formulate their thinking. This strategy supports content vocabulary development and use in appropriate context.

The use of journaling is certainly an appropriate tool, but if the prompt is not clearly stated it may lead to confusion and not promote reflective thinking.

An example of a prompt that is too broad and lacks focus is, “Tell me what you learned today about fractions.” This is too ambiguous for many students, and they may not know where to start. 

Revising the prompt as follows may prove easier for students to comprehend.

Providing a word bank adds another dimension to the reflection process and connects the reflection to the content objective in a much more cohesive manner.

Additionally, emotionally supportive environments in the classroom that encourage re-evaluation of work and conclusions are helpful. I often ask students to “pair-share” and, during the share, instruct them to “revise and edit” as part of the share if new or better ideas come to them. I want the students I work with to understand that the whole point of the sharing is to compare, contrast, challenge, and correct their thinking! We are not looking for a final answer immediately during this phase of instruction.   

Reflective thinking is a life skill!

Our society is becoming more complex. Information is becoming available and changing more rapidly, prompting users to constantly rethink, switch directions, and change problem-solving strategies. Thus, it is increasingly important to prompt reflective thinking during learning to help learners develop strategies to apply new knowledge to the complex situations in their day-to-day activities.

Ideas for creating the right environment for reflection:

• Model metacognitive and self-explanation strategies on specific problems to help students build an integrated understanding of the process of reflection  
• Integrate study guides or advance organizers into classroom materials to prompt students to reflect on their learning
• Use questioning strategies to prompt reflective thinking, specifically getting students to respond to why, how, and what specific decisions are made

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